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A136246
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a(n) = (1/n!)*Sum_{k=0..n} (-1)^(n-k)*Stirling1(n,k)*A062208(k).
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2
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1, 1, 32, 2712, 449102, 122886128, 50225389432, 28670796914144, 21789885975738524, 21271115441652577064, 25938193213744579451420, 38638907727108476424404864, 69044758685363149615280762608, 145768622491129079115419544343808, 358961215083489204505055286181798208
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OFFSET
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0,3
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LINKS
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FORMULA
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a(n) = Sum_{m>=0} binomial(binomial(m,3)+n-1,n)/2^(m+1).
a(n) = Sum_{j=0..3*n} binomial(binomial(j,3)+n-1, n) * (Sum_{i=j..3*n} (-1)^(i-j)*binomial(i,j)). - Andrew Howroyd, Feb 09 2020
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MAPLE
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A000629 := proc(n) local k ; sum( k^n/2^k, k=0..infinity) ; end: A062208 := proc(n) option remember ; local a, stir, ni, n1, n2, n3, stir2, i, j, tmp ; a := 0 ; if n = 0 then RETURN(1) ; fi ; stir := combinat[partition](n) ; stir2 := {} ; for i in stir do if nops(i) <= 3 then tmp := i ; while nops(tmp) < 3 do tmp := [op(tmp), 0] ; od: tmp := combinat[permute](tmp) ; for j in tmp do stir2 := stir2 union { j } ; od: fi ; od: for ni in stir2 do n1 := op(1, ni) ; n2 := op(2, ni) ; n3 := op(3, ni) ; a := a+combinat[multinomial](n, n1, n2, n3)*(A000629(3*n1+2*n2+n3)-1/2-2^(3*n1+2*n2+n3)/4)*(-3)^n2*2^n3 ; od: a/(2*6^n) ; end: A136246 := proc(n) local k ; add((-1)^(n-k)*combinat[stirling1](n, k)*A062208(k), k=0..n)/n! ; end: seq(A136246(n), n=0..14) ; # R. J. Mathar, Apr 01 2008
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MATHEMATICA
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a[n_] := Sum[Binomial[Binomial[j, 3] + n - 1, n] * Sum[(-1)^(i - j)* Binomial[i, j], {i, j, 3n}], {j, 0, 3n}];
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PROG
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(PARI) a(n) = {sum(j=0, 3*n, binomial(binomial(j, 3)+n-1, n) * sum(i=j, 3*n, (-1)^(i-j)*binomial(i, j)))} \\ Andrew Howroyd, Feb 09 2020
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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